Animating slicing up an object
Dissolve slice
Use a Boolean modifier with and animated intersection object:

Chop slice

Add a Boolean modifier to your mesh and set the target boolean object.
Shift+select the boolean object and Object | Parent | Object (or Ctrl+P, o).


You can now use the boolean object to move the mesh without affecting the boolean operation.


With them both still selected Object | Duplicate Linked (or Alt+D), cancel the translation (Esc) and change the boolean operation type of the new mesh to Difference.


You should now have two half's that can be moved separately.


Finally create a shape key on one of the boolean objects that you'll animate to enable/place the boolean/chop.